US 7912656 B2 Abstract A system and method for providing amplitude spectroscopy is provided. Generally, the system contains a generator for providing a waveform for analysis of a multilevel quantum system, wherein the generator has the capability of changing amplitude of the waveform provided and driving the multilevel quantum system at a fixed frequency while sweeping amplitude. A detector is also provided for reading population in different energy states of the multilevel quantum system, wherein the detector plots an amplitude spectroscopy response of the multilevel quantum system. A memory and processor are provided within the system where the processor is configured by the memory to perform the step of plotting an energy-level diagram of the multilevel quantum system from the amplitude spectroscopy plot of the multilevel quantum system.
Claims(21) 1. A system for providing amplitude spectroscopy of a multilevel quantum system, comprising:
a generator for providing a waveform for analysis of a multilevel quantum system, wherein the generator has the capability of changing amplitude of the waveform provided and driving the multilevel quantum system at a fixed frequency while sweeping amplitude;
a detector for reading population in different energy states of the multilevel quantum system, wherein the detector plots an amplitude spectroscopy response of the multilevel quantum system;
a memory; and
a processor configured by the memory to perform the step of plotting an energy-level diagram of the multilevel quantum system from the amplitude spectroscopy plot of the multilevel quantum system.
2. The system of
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8. A system for providing amplitude spectroscopy of a multilevel quantum system, comprising:
logic configured to drive the multilevel quantum system at a fixed frequency, while sweeping amplitude;
logic configured to plot amplitude spectroscopy response of the multilevel quantum system driven toward saturation, where the amplitude spectroscopy response reflects a fixed frequency and a sweeping amplitude; and
logic configured to analyze the amplitude spectroscopy response of the multilevel quantum system to derive an energy level structure of the multilevel quantum system.
9. The system of
10. The system of
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14. A method of providing amplitude spectroscopy of a multilevel quantum system, comprising the steps of:
driving the multilevel quantum system at a fixed frequency, while sweeping amplitude;
determining an amplitude spectroscopy response of the multilevel quantum system, where the amplitude spectroscopy response reflects a fixed frequency and a sweeping amplitude; and
analyzing the amplitude spectroscopy response of the multilevel quantum system to derive an energy level structure of the multilevel quantum system.
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Description This application claims priority to U.S. Provisional Application entitled, “AMPLITUDE SPECTROSCOPY OF A SOLID-STATE ARTIFICIAL ATOM,” having patent application Ser. No. 61/093,907, filed Sep. 3, 2008, which is entirely incorporated herein by reference. This invention was made with government support under Grant Number FA8721-05-C-0002 awarded by the Air Force, grant number F49620-01-1-0457 awarded by the Air Force Office for Sponsored Research, and grant number HR0011-06-C-0051 awarded by the Defense Advanced Research Projects Agency. The government has certain rights in this invention. The present invention is generally related to spectroscopy, and more particularly is related to determining energy spectrums of multi-level quantum systems (including, but not limited to, natural and artificial atoms, molecules, defects, and impurities) by use of amplitude spectroscopy to sweep amplitude and then analyzing results of the amplitude sweeping. Since Newton's dispersion of light into a continuous color “spectrum,” spectroscopy has been viewed primarily as a frequency-based technique. Bunsen, Foucault, Kirchhoff, and many others identified unique spectral lines for elements and compounds based on the emission and absorption of radiation at various frequencies. Spectroscopy has historically been used to obtain a wide range of information about nuclear, atomic, and molecular properties. Early on, the determination of spectral lines, or energy levels, helped elucidate the principles of quantum mechanics through studies of the hydrogen atom and provided a means for testing atomic theory. Since then, several spectroscopy techniques to determine absolute transition frequencies (or, equivalently, wavelengths) have been developed, involving the emission, absorption, or scattering (e.g., Raman) of radiation. The series of spectral lines of hydrogen are named for Balmer and Rydberg, who observed them within and beyond the visible wavelengths. As previously mentioned, such frequency-dependent absorption and emission spectroscopy played a fundamental role in the development of quantum mechanics and the “new” atomic theory by identifying discrete energy levels. With the invention of coherent high-intensity radiation sources at microwave (maser) and optical (laser) frequencies, with tunable, narrow spectral line-widths, targeted absorption spectroscopy of atoms and molecules with high frequency resolution is provided. The advent of tunable, coherent radiation sources at microwave and optical frequencies led to the age of modern atomic spectroscopy, where a primary approach is to identify absorption spectra of natural and artificial atoms and molecules as the source frequency v is varied to fulfill the resonance conditions ΔE=hv, where ΔE is the energy-level separation and h is Planck's constant. The technique is now commonplacein research labs and usually involves shining a beam of light on a sample and watching how it absorbs light as the frequency of the radiation is swept through a range of values. An atom, for example, absorbs radiation at a specific set of frequencies that correspond to gaps between the energy levels of its electrons. Spectroscopy has traditionally been viewed as a frequency-based measurement technique. Frequency-dependent absorption and emission spectroscopy has long played a fundamental role in the characterization of quantum systems. As previously mentioned, the development of coherent microwave (maser) and optical (laser) sources, high-intensity radiation with tunable, narrow spectral line-width, has further enabled targeted absorption spectroscopy of atoms and molecules with high frequency resolution. However, the application of broadband frequency spectroscopy is not universally straightforward. This is particularly relevant for certain classes of multi-level quantum systems (including, but not limited to, natural and artificial atoms, molecules, defects, impurities, which assume quantized energy levels that extend into microwave, millimeter wave and terahertz regimes. Although certainly not an impossible task, a broadband frequency-based spectroscopic study of such multilevel quantum systems in excess of around 50 GHz, becomes extremely challenging and expensive to implement due to numerous frequency-dependent effects (e.g., frequency dispersion and the requisite tolerances to control impedance), and due to the general requirement of multipliers that are inefficient and intrinsically noisy. The abovementioned difficulty has been problematic for researchers performing studies on multilevel quantum systems, such as, for example, artificial atoms. Artificial atoms exhibit properties of ordinary atoms, including discrete energy levels. Such atoms could potentially be used to store and process data. Unfortunately, the problem in using artificial atoms as putative quantum-information systems is that the gaps between the levels tend to be in the problematic millimeter and microwave region. Thus, a heretofore unaddressed need exists in the industry to address the aforementioned deficiencies and inadequacies. Embodiments of the present invention provide a system and method for providing amplitude spectroscopy of a multilevel quantum system. Briefly described, in architecture, one embodiment of the system, among others, can be implemented as follows. The system contains: a generator for providing a waveform for analysis of a multilevel quantum system, wherein the generator has the capability of changing amplitude of the waveform provided and driving the multilevel quantum system at a fixed frequency while sweeping amplitude; a detector for reading population in different energy states of the multilevel quantum system, wherein the detector plots an amplitude spectroscopy response of the multilevel quantum system; a memory; and a processor configured by the memory to perform the step of plotting an energy-level diagram of the multilevel quantum system from the amplitude spectroscopy plot of the multilevel quantum system. The present invention can also be viewed as providing methods for providing for providing amplitude spectroscopy of a multilevel quantum system. In this regard, one embodiment of such a method, among others, can be broadly summarized by the following steps: driving the multilevel quantum system at a fixed frequency, while sweeping amplitude; determining an amplitude spectroscopy response of the multilevel quantum system driven toward saturation, where the amplitude spectroscopy response reflects a fixed frequency and a sweeping amplitude; and analyzing the amplitude spectroscopy response of the multilevel quantum system to derive an energy level structure of the multilevel quantum system. Other systems, methods, and features of the present invention will be or become apparent to one with skill in the art upon examination of the following drawings and detailed description. It is intended that all such additional systems, methods, and features be included within this description, be within the scope of the present invention, and be protected by the accompanying claims. Many aspects of the invention can be better understood with reference to the following drawings. The components in the drawings are not necessarily to scale, emphasis instead being placed upon clearly illustrating the principles of the present invention. Moreover, in the drawings, like reference numerals designate corresponding parts throughout the several views. The present system and method provides for use of amplitude spectroscopy. Amplitude spectroscopy is a technique that allows broadband spectroscopic characterization of a multilevel quantum system. With amplitude spectroscopy, spectroscopic information is obtained from the system response to driving field amplitude at a fixed frequency. The resulting spectroscopic interference patterns, referred to herein as “spectroscopy diamonds,” or “diamonds”, are mediated by multilevel Landau-Zener-Stückelberg (LZS) transitions and Mach-Zehnder-type interferometry, and they serve as a fingerprint of the multilevel energy spectrum of an atom. The energy spectrum is then determined by analyzing the atomic fingerprint. In this way, the amplitude spectroscopy technique complements frequency spectroscopy. Although a less direct approach, amplitude spectroscopy allows one to probe the energy level structure of a quantum system over extraordinarily large (even practically prohibitive) bandwidths by circumventing many of the challenges associated with a frequency-based approach. It should be noted that although the present description refers to “artificial atoms” at times, the present system and method can be used in principle to identify the energy level structure (and any subsequent classification that this knowledge enables) of any atom, molecule, defect, or impurity with multiple energy levels that exhibit avoided crossings, either intrinsically or in the presence of a driving field, and whose quantum state can be driven by the application of a driving field towards, away, and/or through those level crossings. Specifically, although but one example of the broader applicability of amplitude spectroscopy, an artificial atom is used to demonstrate amplitude spectroscopy of a multilevel quantum system. The generator A detector Readings from the detector are forwarded to a computer Functionality for using the results of the detector Generally, in terms of hardware architecture, as shown in The processor The memory The software The system The I/O devices When the system When the system The computer readable medium can be, for example but not limited to, an electronic, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus, device, or propagation medium. More specific examples (a nonexhaustive list) of the computer-readable medium would include the following: an electrical connection (electronic) having one or more wires, a portable computer diskette (magnetic), a random access memory (RAM) (electronic), a read-only memory (ROM) (electronic), an erasable programmable read-only memory (EPROM, EEPROM, or Flash memory) (electronic), an optical fiber (optical), and a portable compact disc read-only memory (CDROM) (optical). Note that the computer-readable medium could even be paper or another suitable medium upon which the program is printed, as the program can be electronically captured, via for instance optical scanning of the paper or other medium, then compiled, interpreted or otherwise processed in a suitable manner if necessary, and then stored in a computer memory. In an alternative embodiment, where the system Having generally described the devices in the present system and method, the following further describes the processes used in the present system and method for providing amplitude spectroscopy of an atom. As is shown by block To reach a regime dominated by Landau-Zener transitions at level crossings, which is necessary to determine locations of energy levels, the driving frequency, represented as v=ω/2π, is chosen such that hv is generally much smaller that the instantaneous energy-level spacing throughout the driving cycle, but the evolution through level crossings is non-adiabatic. The transition rate between the states q and q′ is controlled by the relative-energy sweep rate represented by equation 2 (Eq. 2). Repeated Landau-Zener transitions give rise to Stückelberg oscillations in the populations of the states q and q′. For a crossing Δ For exemplary purposes, the pulse sequence shown in Returning to
Returning to For the particular static flux detuning δf _{1}, where the Δ_{0,0 }crossing is reached (1; other level-crossing voltages are similarly labeled V_{2}, . . . , V_{5}). For V_{1}<V<V_{2}, Stüickelberg interference at the Δ_{0,0 }crossing results in the observed fringe contrast of 1 and D2 interference patterns due to a single (D1) and multiple (D2) avoided crossings. It is noted that there is strong population inversion in D2 and that there is cooling in the region between D1 and D2 as well as at interference nodes in D2. Arrows indicate the locations δf_{q,q′} of avoided crossings, as shown on the top axis.
Referring to both 1. For V_{2}<V<V_{3}, the data show a large reduction in contrast due to the addition of a single, strong transition at Δ_{1,0}. The saturated population depends on the competition between transitions at Δ_{0,0 }and Δ_{1,0}, on relatively fast intra-well relaxation and to a lesser extent, on much slower inter-well relaxation processes. In the qubit 30, because Δ_{0,0}<<Δ_{1,0}, the dominant transitions occur at the Δ_{1,0 }crossing. Transitions |0,L→|0,R are still induced at the Δ_{0,0 }crossing, but constructive Stückelberg interference at Δ_{1,0 }converts a substantial fraction of that population to |1, L, an excited state of the left-hand well. Since relaxation within a well is a relatively fast process in this qubit in comparison with the relaxation between wells, the excited state population tends to relax back to the ground state, |0, L. thus suppressing the net population transfer. In contrast, for values of V such that the interference at Δ_{1,0 }is destructive, the population remains in |0, R, making the interference fringes arising from Δ_{0,0 }visible, albeit with reduced contrast (_{2}<V<V_{3}, the qubit can be cooled to its ground state.
At even larger amplitudes, transitions to additional excited states become possible. For V>V 2. The right-hand side of D2 is marked by the amplitude, V=V_{4}, where Δ_{2,0 }is reached, allowing transitions between |0,R and |2,R. This description can be extended straightforwardly to the remainder of the spectrum, and therefore further explanation is not provided herein.
There are several notable features associated with amplitude spectroscopy, as provided by the present invention. First, the qubit Third, the second diamond D _{1,0 }and Δ_{0,1 }combined with fast intrawell relaxation to |0,L and |0,R (_{1,0 }and Δ_{0,1 }have strong oscillatory behavior due to Stückelberg interference, which is constructive or destructive depending on the values of δf_{dc }and V. The competition between these rates leads to the observed checkerboard pattern, symmetric about δf_{dc}=0, with alternating regions of strong population inversion and efficient cooling, depending on the specific well (left-hand well or right-hand well) in which the relaxation occurs. Similar checkerboard patterns are present in the diamonds D3 and D4. The population inversion observed here can be used as the active medium of a single-or multi-atom laser.
As previously mentioned, many different techniques may be used to derive the energy level diagram from the amplitude spectroscopy plot of the driven qubit. As an example, energy level slopes may be determined. Energy level slopes can be derived from fitting the diamond (Stückelberg interference) patterns from the response of the atom driven for long times (driven into saturation). Energy level slopes can also be derived from the 2D Fourier transform of the amplitude response, or from the diamond (Stückelberg interference) patterns from the response of the atom driven for short times. Avoided crossing values may also be determined. As an example, avoided crossing values (splittings) can be derived from fitting the diamond (Stückelberg interference) patterns from the response of the atom driven for long times (driven into saturation). In addition, avoided crossing values (splittings) can be derived from fitting the diamond (Stückelberg interference) patterns from the response of the qubit driven for short times. Positions of avoided crossing values may also be determined to assist in deriving the energy level diagram from the amplitude spectroscopy plot of the driven qubit. As an example, the vertices of the diamonds provide the flux positions of the avoided crossings. Equivalently, one can extrapolate the diamond edges back to the flux axis (vertical axis on diamond plots) in order to find their location in flux. The following further describes certain of these techniques for deriving the energy level diagram from the amplitude spectroscopy plot of the driven qubit. The energy-level separation ΔE _{q,q′} of the avoided crossing Δ_{q,q′} and to the sum of the magnitudes of the energy-level slopes m_{q }and m_{q′}. Since the relative phase accumulated between the |q,L and |q′,R components of the wavefunction over repeated Landau-Zener transitions is sensitive to ΔE_{q,q′}, the slopes can be derived from the interference patterns, which arise when δf_{dc }is varied. The Nth node in the interference pattern, where a ‘node’ indicates a minimal change in the states' populations, occurs when a relative phase or 2πN is accumulated between transitions. For sinusoidal driving, the locations of the nodes (in δf_{dc}) follow the power law s_{q,q′}N^{2/3}, with a prefactor s_{q,q′} related to the energy-level slopes by equation 4 below
As an alternative way to analyze the data, as mentioned above, one can use the discrete two-dimensional Fourier transform (2DFT). To see the benefits of the 2DFT, it is noted that the amplitude spectroscopy plots in In particular, the 2DFT allows for determining of the relation between the slopes m The amplitude spectroscopy data typically exhibit complex checkerboard patterns of Stückelberg oscillations originating from several level crossings, superimposed with multi-photon resonance structure on a finer scale. Extracting spectroscopic information from these structures is facilitated by the 2DFT, which yields Fourier intensity localized near lemon-shaped oval curves in Fourier space. The following describes how these results can be used to extract the parameters of the qubit energy spectrum from the data. It should be noted that this is merely provided as an example. We start with the first diamond (D The 2DFT of the transition rate displays intensity concentrated along the two curves as shown by equation 5 Most strikingly, the apparently distinct phenomena of interference fringes and multi-photon resonances observed in the real space image are manifested as a single smooth curve in Fourier space. This structure can be understood by considering k The situation in the second diamond is somewhat more complicated. In numerical simulations it is noted that the steady-state magnetization in D
The sinusoid has the period different from that of equation 7 by the ratio of the slopes 2|m Another important feature of 2DFT of the higher diamonds D Taking the second diamond D
Using the formulas of equation 11, these restrictions can be more conveniently expressed as −k
It is noted that the actual size of the arcs visible in 2DFT of the second diamond D The restriction of the phase space sampled by 2DFT becomes even more stringent for higher diamonds D Slope Extraction from Landau-Zener-Stückelberg Interference Patterns The following technique provides an example of one way to determine the energy band slopes. The interference between sequential Landau-Zener transitions at an isolated avoided crossing is sensitive to the relative phase shown by equation 15 below accumulated by the two components of the wave function between the first and second traversals of the avoided crossing. For demonstration, we focus on the interference in the first diamond D Δ E(t)≈2|m _{0}|(αV−δf _{dc})−|m _{0}|αVω^{2}(t−T/4)^{2} (Eq. 17)In equation 17, T=2π/ω is the period of the driving signal, and m _{0 }is the energy-level slope of the ground state (assumed to be equal in magnitude for the left and right wells).
By setting ΔE(t*)=0, it is noted that the initial and final crossing times t
Using the quantization condition for interference, Δφ=2πN, and the definition of t*, we find the values of static flux detuning {δf
Equation 20 can be generalized to any avoided crossing Δ Fresnel-like Oscillations in the Landau-Zener Dynamics Information may also be extracted from the application of amplitude spectroscopy over short time scales, in addition to saturated driving. The time-dependent oscillations observed in temporal-response measurements result from Larmor precession about a tilted axis following the qubit's transit through an avoided crossing. In the regime where the Landau-Zener transition probability is small, we use a perturbative model to relate these oscillations to the well-known Fresnel integral. By linearizing the sinusoidal driving signal δf(t)=−δf t{circumflex over (σ)} ^{z}+Δ{circumflex over (σ)}^{x}), β=A_{q,q′}Φ_{rf}ω cos ωt, (Eq. 21)where β is the sweep velocity, the detuning δf _{dc }is measured from the level crossing, Δ=Δ_{q,q′} is the energy splitting, and A_{q,q′}=h(|m_{q}|+|m_{q′}|) is the energy-flux conversion factor. Next, we transform to a non-uniformly rotating frame by |ψ_{R}(t)=e^{iφ(t){circumflex over (σ)}} ^{ z }|ψ(t)with
We the expand the system's time evolution operator U(t, t _{0})|↓|^{2 }to find the system in the state |↓ at time t given that it started in the state |↓ at t0′-′/β is given by equation 24 below.
Detailed fitting of the observed oscillations for the states |0,L and |2,R requires simulating the full Bloch dynamics with the sinusoidal driving and smooth pulse turn-off after the RF pulse ends at t=Δt, as well as taking into account intrawell relaxation after the LZ transition. For that we use a non-Hermitian Hamiltonian, as shown by equation 25, below. For exemplary purposes, we estimate Δ Gap and Slope Extraction from Amplitude Spectroscopy Over Short Time Scales The temporal oscillations, or ‘ringing’, can be understood qualitatively in a pseudo-spin ½ A picture, in which the qubit states are identified with up-and down-spin states relative to a fictitious z axis. The qubit To obtain a quantitative fit we account for decoherence and the non-abrupt ending of the pulse, which adds a small Stückelberg-type interference contribution. We find good agreement between the data and a simulation of the Bloch dynamics of the two-level system near Δ As in the case of the stationary driving, the energy-level slopes can be extracted from the Stückelberg fringes using the N Returning to It should be emphasized that the above-described embodiments of the present invention are merely possible examples of implementations, merely set forth for a clear understanding of the principles of the invention. Many variations and modifications may be made to the above-described embodiments of the invention without departing substantially from the spirit and principles of the invention. All such modifications and variations are intended to be included herein within the scope of this disclosure and the present invention and protected by the following claims. Patent Citations
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